Finite Group Databasebeta

← #20277927

Subgroups — #20277927

Order 6784.

944 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 167 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#11719947C211yesmaximal, characteristic, complemented
2C2×D4×Dic53C212yesmaximal, complemented
3C2×#13133177C211yesmaximal, characteristic, complemented
4#16810544C211yesmaximal, characteristic, complemented
5#16810539C212yesmaximal, complemented
6#16810537C211yesmaximal, characteristic, complemented
7#16810535C211yesmaximal, characteristic, complemented
8#16810531C212yesmaximal, complemented
9C22⋊C4×Dic53C212yesmaximal, complemented
10#16810470C211yesmaximal, characteristic, complemented
11C53×#392C211yesmaximal, characteristic, complemented
12C2×#11719947—28no—
13C2×#11719947—24no—
14C22×Dic106C2211yescharacteristic
15C22×C22⋊D53C2211yescharacteristic
16C22×C22⋊D53C2212yes—
17C23×Dic53C2211yescharacteristic
18C23×Dic53C2212yes—
19C22×C4×D53C2211yescharacteristic
20D4×Dic53—28no—
21#13133177—24no—
22C2×C4⋊Dic53C2212yes—
23C2×#11719919C2211yescharacteristic
24C2×C22⋊Dic53C2212yes—
25C2×C22⋊Dic53C2212yes—
26C2×C22⋊Dic53C2212yes—
27C2×#11719921C2212yes—
28C2×#11719921C2211yescharacteristic
29C2×C4×Dic53C2211yescharacteristic
30C2×C4×Dic53C2212yes—
31#13133124C2212yes—
32#13133124C2211yescharacteristic
33#13133124C2212yes—
34C22×D4×C53C2212yes—
35C22×D4×C53C2211yescharacteristic
36C2×C22⋊C4×C53C2211yescharacteristic
37C2×C22⋊C4×C53C2212yes—
38#95×C53C2211yescharacteristic
39C24×C53C2312yes—
40C24×C53C2311yescharacteristic
41C23×D53C2311yescharacteristic
42C2×D4×C53—24no—
43C2×D4×C53—24no—
44C2×D4×C53—24no—
45C2×D4×C53—22no—
46C2×D4×C53—24no—
47C22×C4×C53C2311yescharacteristic
48C22×C4×C53C2311yescharacteristic
49C22×C4×C53C2312yes—
50C2×C22⋊D53—24no—
51C2×C22⋊D53—22no—
52C2×C22⋊D53—22no—
53C2×C22⋊D53—24no—
54C2×C22⋊D53—24no—
55C2×C22⋊D53—22no—
56C22×Dic53—24no—
57C22×Dic53—24no—
58C22×Dic53—22no—
59C22×Dic53—24no—
60C22×Dic53D418yescomplemented
61C22×Dic53C2312yes—
62C22×Dic53C2311yescharacteristic
63C22×Dic53C2312yes—
64C22×Dic53C2311yescharacteristic
65C22×Dic53C2311yescharacteristic
66#11719947—232no—
67C2×C4×D53—24no—
68C2×C4×D53D414yescomplemented
69C2×Dic106—22no—
70C2×Dic106—24no—
71C22⋊C4×C53—22no—
72C22⋊C4×C53—24no—
73C22⋊Dic53—24no—
74C22⋊Dic53—24no—
75C22⋊Dic53—24no—
76#11719921—22no—
77#11719921—24no—
78C4⋊Dic53—24no—
79#11719919—22no—
80C4×Dic53—22no—
81C4×Dic53—24no—
82C23×C53—22no—
83C23×C53—22no—
84C23×C53—21no—
85C23×C53—22no—
86C23×C53—21no—
87C23×C53—22no—
88C23×C53—22no—
89C23×C53—22no—
90C23×C53—22no—
91C23×C53—22no—
92C23×C53C2×D414yes—
93C23×C53#4112yescomplemented
94C23×C53C2411yescharacteristic
95C22×D53—22no—
96C22×D53—21no—
97C22×D53—21no—
98C22×D53—22no—
99C22×D53C2×D412yes—
100D4×C53—28no—
101D4×C53—216no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53—21no—
106C2×C4×C53—22no—
107C2×C4×C53C2×D412yes—
108C2×C4×C53#4114yescomplemented
109C22⋊D53—216no—
110C22⋊D53—28no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—216no—
116C2×Dic53—21no—
117C2×Dic53—22no—
118C2×Dic53—22no—
119C2×Dic53—22no—
120C2×Dic53—22no—
121C2×Dic53—44no—
122C2×Dic53C2×D414yes—
123C2×Dic53C2×D414yes—
124C2×Dic53C2×D412yes—
125C2×Dic53#4112yescomplemented
126C4×D53—28no—
127Dic106—28no—
128C22×C53—24no—
129C22×C53—22no—
130C22×C53—24no—
131C22×C53—24no—
132C22×C53—24no—
133C22×C53—41no—
134C22×C53—42no—
135C22×C53—42no—
136C22×C53—42no—
137C22×C53—42no—
138C22×C53—42no—
139C22×C53—41no—
140C22×C53C22×D411yescharacteristic
141C22×C53C22×D412yes—
142C22×C53C2×#4111yescharacteristic
143C22×C53C2×#4111yescharacteristic
144C22×C53C2×#4112yes—
145C2×D53—22no—
146C2×D53—22no—
147C2×D53—22no—
148C2×D53—42no—
149C2×D53—42no—
150C4×C53—22no—
151C4×C53—24no—
152C4×C53—41no—
153Dic53—24no—
154Dic53—22no—
155Dic53—41no—
156Dic53—42no—
157Dic53#12014yes—
158#2388—531nomaximal, Sylow
159C2×C53—44no—
160C2×C53—42no—
161C2×C53C2×#12011yescharacteristic
162C2×C53#52512yes—
163C2×C53#52511yescharacteristic
164C2×C53#53212yes—
165C2×C53#53211yescharacteristic
166D53—42no—
167C22×#41—531no—
168C2×C4×D4—532no—
169C2×C4×D4—531no—
170#395—531no—
171#394—531no—
172#394—532no—
173#393—531no—
174#392—532no—
175#392D5311yescharacteristic, complemented
176C4×C22⋊C4—532no—
177C4×C22⋊C4—531no—
178C53#238811yescharacteristic, complemented, Sylow
179C2×#41—1068no—
180C2×#41—1064no—
181C22×Q8—531no—
182C22×D4—531no—
183C22×D4—532no—
184C22×D4C2×D5312yes—
185C22×D4C2×D5311yescharacteristic
186C23×C4—532no—
187C23×C4—531no—
188C23×C4—531no—
189C4×D4—1064no—
190C4×D4—1068no—
191C2×C4⋊C4—531no—
192C2×C4⋊C4—532no—
193C2×C22⋊C4—532no—
194C2×C22⋊C4—532no—
195C2×C22⋊C4—532no—
196C2×C22⋊C4—531no—
197C2×C22⋊C4—532no—
198C2×C22⋊C4C2×D5311yescharacteristic
199C2×C22⋊C4C2×D5312yes—
200C2×C42—531no—
201C2×C42—532no—
202#95—532no—
203#95—532no—
204#95—531no—
205#95C2×D5311yescharacteristic
206C24—531no—
207C24C22×D5312yes—
208C24C22×D5311yescharacteristic
209#41—10632no—
210C2×Q8—1064no—
211C2×Q8—1062no—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—24no—
215C2×D4—22no—
216C2×D4—24no—
217C2×D4—1062no—
218C2×D4—1064no—
219C2×D4—1062no—
220C2×D4—1064no—
221C2×D4—1064no—
222C2×D4—1062no—
223C22×C4—531no—
224C22×C4—538no—
225C22×C4—531no—
226C22×C4—532no—
227C22×C4—534no—
228C22×C4—532no—
229C22×C4—531no—
230C22×C4—1064no—
231C22×C4—1064no—
232C22×C4—1064no—
233C22×C4—1062no—
234C22×C4—1064no—
235C22×C4C22×D5311yescharacteristic
236C22×C4C22×D5311yescharacteristic
237C22×C4C22×D5312yes—
238C4⋊C4—1064no—
239C4⋊C4—1062no—
240C22⋊C4—22no—
241C22⋊C4—24no—
242C22⋊C4—1062no—
243C22⋊C4—1064no—
244C22⋊C4—1064no—
245C22⋊C4—1064no—
246C22⋊C4—1064no—
247C42—1064no—
248C42—1062no—
249C23—21no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—21no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—532no—
260C23—1062no—
261C23—1061no—
262C23—1062no—
263C23—1061no—
264C23D4×D5314yes—
265C23#1171994712yescomplemented
266C23C23×D5311yescharacteristic
267Q8—1068no—
268D4—216no—
269D4—28no—
270D4—1068no—
271D4—10616no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—534no—
278C2×C4—532no—
279C2×C4—534no—
280C2×C4—532no—
281C2×C4—1068no—
282C2×C4—10616no—
283C2×C4—1061no—
284C2×C4—1062no—
285C2×C4—1062no—
286C2×C4—1061no—
287C2×C4—1062no—
288C2×C4—1062no—
289C2×C4—1061no—
290C2×C4—1062no—
291C2×C4—1062no—
292C2×C4—2124no—
293C2×C4D4×D5312yes—
294C2×C4#1171994714yescomplemented
295C22—24no—
296C22—24no—
297C22—24no—
298C22—22no—
299C22—24no—
300C22—41no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—41no—
306C22—42no—
307C22—1062no—
308C22—1062no—
309C22—1062no—
310C22—2122no—
311C22—2122no—
312C22C2×D4×D5312yes—
313C22C2×D4×D5311yescharacteristic
314C22#41×D5311yescharacteristic
315C22C2×#1171994711yescharacteristic
316C22C2×#1171994712yes—
317C4—22no—
318C4—24no—
319C4—41no—
320C4—534no—
321C4—1064no—
322C4—1062no—
323C4—2121no—
324C4—2122no—
325C2—44no—
326C2—42no—
327C2—2122no—
328C2#120×D5311yescharacteristic
329C2#1681112912yes—
330C2#1681113311yescharacteristic
331C2#1681116812yes—
332C2#1681117111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#392C2
hypercenter#392D53
hypocenterC53#2388
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #392

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×#11719947C2semidirect
2. C2×D4×Dic53C2semidirect
3. C2×#13133177C2semidirect
4. #16810544C2semidirect
5. #16810539C2semidirect
6. #16810537C2semidirect
7. #16810535C2semidirect
8. #16810531C2semidirect
9. C22⋊C4×Dic53C2semidirect
10. #16810470C2semidirect
11. C53×#392C2semidirect
60. C22×Dic53D4semidirect
68. C2×C4×D53D4semidirect
93. C23×C53#41semidirect
108. C2×C4×C53#41semidirect
125. C2×Dic53#41semidirect
175. #392D53semidirect
178. C53#2388semidirect
265. C23#11719947semidirect
294. C2×C4#11719947semidirect